W erner’s m odel
5.5 Introducing w ind speedup
W ind speedup has been noted, bo th in th e field and in th e wind tunnel exper im ents (see for example M ulligan, 1988; Frank and K ocurek, 1996b; Lancaster
et al, 1996; Wiggs et al, 1996; see also section 2.4.4). M ulligan found th a t the wind speed increases linearly over a transverse dune. T he m ore im p o rtan t quan tity here is wind shear velocity (u*), or equivalently the w ind shear stress, th a t causes sand tra n sp o rt. Lancaster and co-workers found th a t th e shear velocity increases linearly over a barchan dune, provided th a t the w ind is strong enough such th a t th e shear velocity significantly exceeds th e threshold shear velocity (L ancaster et al, figure 10). This wind speedup over a w indw ard slope of a dune is necessary to sustain the shape of th e windw ard slope (Bagnold, 1941; Zeman and Jensen, 1988; see also section 3.2.1).
In a revised model, Nishimori and co-workers a tte m p ted to introduce a more realistic slab tra n sp o rt rule com pared to W erner’s approach, based on th e 1- dim ensional wind fiow calculation (2-dim ensional wind fiow p a tte rn ) over a well- developed dune, in which wind speedup is included (Nishimori et al, 1998; section 2.7.2). In this section, a wind speedup effect is introduced into W erner’s model w ith a kinem atic rule.
5.5.1
Linear wind speedup: K inem atic form ulation
If dunes m igrate downwind a t a constant speed (cd) w ithout changing their shapes as is widely believed (Bagnold, 1941; section 2.4.6), then looking back to section 2.4.1 th e following equation (2.2) holds (for derivation, see section 3.2.1):
drj dq
where 7 is th e sand bulk density in dunes, r){x, t) is th e local height and q is the sand flux. Integrating (2.2) from x = — 0 0 to x = x leads to
q{x) - q {-o o) = 7Cd r]{x). (5.3) Let us assume th a t the sand flux {q{x)) obeys B agnold’s sand tra n sp o rt form ula (2.5):
q{x) = Cb ( ^ ) (w*(^))^,
- ^ r 9
where Cb is a constant, Dg is sand grain diam eter, is reference diam eter of 0.25 m m , is air density and g is gravitational acceleration. A ssum ing th a t the
shear velocity on a dune surface is 6u^(x) larger th a n th a t on a bare ground, due to th e wind speedup over the dune, th e corresponding increase of sand flux is w ritten as
— q{—^ ) = C'a (7^)^^^ (— ) [u*(a;)^ — u*(—00)^] -^r 9
= Cb (t^)^'^^ (— ) [(u*(—00) + <^u*(rc))^ — u*(—00)^]
■L-'x 9 V, 00 j
(5.4) If th e increase of shear velocity is small enough:
5u^[x) « u*(—00), equation (5.4) becomes 9(3;) - g (- o o ) = Cb ( i f ( “ ) ^ * ( -00)^ [(1 + 3- ( * ^ \ ) - 1] JJr g u^[ — oo) % S Cb { ^ Y ^ ^ {— ) u ^ { - o o ) ‘^ Su^{x). (5.5) 9
Combining (5.3) and (5.5)
?Cd
3 Cb( ^ ) V 2 ( ^ ) „ . ( _ ^ ) 2
~ — :— - nix)- (5.6)
T hough in W erner’s model, bo th the tran sp o rt length of sand slabs (L) and the num ber of slabs to be removed [5h) are constant (typically L = 5 and 6h = 1),
L is assum ed here to be proportional to it*(a;), so th a t from equation (5.6), L(a;) = Lo + Cl h(a;), (5.7) where h{x) is th e num ber of slabs a t x. Given th a t th e sand flux (g(a;)) can be approxim ated as
q{x) = L(x)
where the coefficient 1/3 is the slab aspect ratio (Nishimori et ai, 1998), since , . _ ôh{x) , -3 ç (-o o ) = Lo % u ^ ( - o o ) , and Sq{x) = SL{x) % u*(—oo)^ ôu^{x), ôh{x) m ust be given by 0h(x) % it*(—oo)^ = const. (5.8) Consequently from equations (5.7) and (5.8), th e wind speedup over a dune can be introduced by letting the tra n sp o rt length of sand slab (L) be
L{x) = Lq + C l {h{x) — h r e f ), (5.9)
where x is th e erosion/ongoing ‘bounce’ site, C i is co n stan t and href is a reference num ber of slabs, which is slightly smaller th a n th e average num ber of slabs (havg)- The exact deflnition of href will be introduced in th e next section (5.6). E quation
(5.9) can be interpreted as the linear shear velocity increase observed by Lancaster
5.5.2 N on-linear wind speedup
In th e previous subsection, by assum ing th a t th e increase of shear velocity is small enough (Su^{x) « u*(—oo)), linear slab tra n sp o rt was discussed. In the field, however, this condition is not always satisfied. A non-linear increase of shear velocity was observed by Frank and K ocurek (1996b, figure 3). More im portantly, saltatio n length, which corresponds to the slab tra n sp o rt length (L) in th e model, non-linearly increases as shear velocity (u*) increases. From th eir m i croscale analysis of saltatin g sand grains, A nderson and H allet established a set of equations describing saltatio n length (A) against shear velocity (u*) (Anderson and H allet, 1986; Anderson, 1988). Figure 5.4 shows th e relation between the saltatio n length (A) and the shear velocity (u*), calculated using th eir equations (2) to (5b) in th e 1988 paper. O ther param eters used here are sand grain diam eter (Dg) of 0.25 m m and lift-off angle of 50° (W hite and Schulz, 1977). Considering these two non-linear relations about shear velocity (u*), it seems to be reasonable th a t a non-linear shear-velocity-increase term (C2) m ay be introduced into slab tra n sp o rt length (L) in the model, so th a t equation (5.9) can be revised as
L { h j ) = Lo + Cl { h{ i , j ) — href) + C2 — href)^ > href)
= Lo + Ci (h(i, j ) - href) (L(2, j ) < href),
(5.10) where Lo, C% and C2 are constants, (i , j ) is th e erosion/ongoing ‘bounce’ site and href is th e reference num ber of slabs defined in th e next section.